Question

Find the indicated term of each geometric sequence. 15th term of \(1,-1,1, \ldots\)

Short answer

The 15th term is 1.

Step-by-step solution

Identify the first term (a)

In the geometric sequence, the first term is given as 1. So, let the first term be denoted as \(a = 1\).

Determine the common ratio (r)

The common ratio \(r\) can be found by dividing any term by the previous term. Here, the second term is -1 and the first term is 1. So, \(r = \frac{-1}{1} = -1\).

Use the general formula for the nth term

The formula for the nth term of a geometric sequence is \(a_n = a \, r^{n-1}\).

Substitute the known values

To find the 15th term, substitute \(a = 1\), \(r = -1\), and \(n = 15\) into the formula: \(a_{15} = 1 \, (-1)^{15-1}\).

Simplify the expression

Simplify the expression: \((-1)^{15-1} = (-1)^{14} = 1\). Therefore, \(a_{15} = 1 \, \times 1 = 1\).

Key concepts

First Term

The first term of a geometric sequence is essentially where our pattern begins. It’s the initial value from which all other terms derive. In your exercise, the first term is clearly given as 1.

In general terms, we denote the first term by the letter \(a\). So, in this case, \(a = 1\). Knowing the first term is crucial because it sets the foundation for our pattern, influencing every other term in the sequence.

Whenever looking for any term in a geometric sequence, always begin by identifying the first term clearly.

Common Ratio

To continue building the sequence after the first term, we need to determine the common ratio. The common ratio (denoted as \(r\)) is the factor by which we multiply each term to get the next term.

This ratio is found by dividing any term by its preceding term. For example, in your exercise:

• The second term is -1.
• The first term is 1.

Thus, the common ratio \(r = \frac{-1}{1} = -1\).

The common ratio is a foundational component because it highlights the pattern of growth or decay in the sequence.
Always check at least two terms to verify your common ratio, ensuring that it applies throughout the sequence.

Nth Term Formula

Once we know the first term and the common ratio, we can find any term in the sequence using the nth term formula. The general formula for the nth term \(a_n\) of a geometric sequence is:
\[a_n = a \, r^{n-1}\]

Here, \(a\) is the first term, \(r\) is the common ratio, and \(n\) is the position of the term in the sequence. For the given exercise, to find the 15th term, we use:
\[a_{15} = 1 \, (-1)^{15-1}\]

Always substitute the known values carefully to avoid errors. The formula gives you precise control over finding terms at any position.

Sequence Simplification

After substituting the values into the nth term formula, we move to simplify the expression. In this case, we need to simplify:
\[(-1)^{14} = 1\]

Since any even power of -1 will be 1, it simplifies our calculation significantly.

Finally, multiplying by the first term, we get:
\[a_{15} = 1 \, \times 1 = 1\]

Simplification is vital because it reduces complexity and makes calculations manageable. Always perform simplification step-by-step.
Remember to apply rules of exponents attentively, especially when dealing with negative bases.